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20
Dualizing complexes and perverse sheaves on noncommutative ringed schemes
, 2002
"... Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to sh ..."
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Cited by 13 (6 self)
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Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to show that a rigid dualizing complex exists on every affine open set in X. In the geometric part of the proof we construct a perverse tstructure on the derived category of bimodules, and this allows us to glue the affine rigid dualizing complexes to
DUALISING COMPLEXES AND TWISTED HOCHSCHILD (CO)HOMOLOGY FOR NOETHERIAN HOPF ALGEBRAS
, 2006
"... Abstract. We show that many noetherian Hopf algebras A have a rigid dualising complex R with R ∼ = ν A 1 [d]. Here, d is the injective dimension of the algebra and ν is a certain kalgebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is her ..."
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Cited by 12 (1 self)
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Abstract. We show that many noetherian Hopf algebras A have a rigid dualising complex R with R ∼ = ν A 1 [d]. Here, d is the injective dimension of the algebra and ν is a certain kalgebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is hereby generalised we call ν the Nakayama automorphism of A. We prove that ν = S 2 ξ, where S is the antipode of A and ξ is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be nonzero in the top dimension d, when A is an ArtinSchelter regular noetherian Hopf algebra of global dimension d. (Twisted) Poincaré duality holds in this setting, as is deduced from a theorem of Van den Bergh. Calculating ν for A using also the opposite coalgebra structure, we determine a formula for S 4 generalising a 1976 formula of Radford for A finite dimensional. Applications of the results to the cases where A is PI, an enveloping algebra, a quantum group, a quantised function algebra and a group algebra are outlined.
Derived Picard groups of finite dimensional hereditary algebras
"... Abstract. Let A be an algebra over a field k, and denote by D b (Mod A) the bounded derived category of left Amodules. The derived Picard group DPick(A) is the group of triangle autoequivalences of D b (Mod A) induced by tilting complexes. In [Ye2] we proved that DPick(A) parameterizes the isomorp ..."
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Cited by 11 (6 self)
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Abstract. Let A be an algebra over a field k, and denote by D b (Mod A) the bounded derived category of left Amodules. The derived Picard group DPick(A) is the group of triangle autoequivalences of D b (Mod A) induced by tilting complexes. In [Ye2] we proved that DPick(A) parameterizes the isomorphism classes of dualizing complexes over A. Also when A is either commutative or local, DPick(A) ∼ = Pick(A) × Z, where Pick(A) is the noncommutative Picard group (the group of Morita equivalences). In this paper we study the group DPick(A) when A = k⃗ ∆ is the path algebra of a finite quiver ⃗ ∆. We obtain general results on the structure of DPick(A), as well as explicit calculations for the Dynkin and affine quivers, and for some wild quivers with multiple arrows. Our method is to construct a representation of DPick(A) on a certain infinite quiver. This representation is faithful when ⃗ ∆ is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor
Dualizing complexes and perverse modules over differential algebras
 COMPOSITIO MATH
, 2004
"... A differential algebra of finite type over a field k is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated kalgebra. The prototypical example is the algebra of differential operators on a smooth affine variety, when char ..."
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Cited by 10 (5 self)
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A differential algebra of finite type over a field k is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated kalgebra. The prototypical example is the algebra of differential operators on a smooth affine variety, when char k = 0. We study homological and geometric properties of differential algebras of finite type. The main results concern the rigid dualizing complex over such an algebra A: its existence, structure and variance properties. We also define and study perverse Amodules, and show how they are related to the Auslander property of the rigid dualizing complex of A.
The homotopy category of complexes of projective modules
, 2003
"... Abstract. The homotopy category of complexes of projective leftmodules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite derived ..."
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Cited by 9 (1 self)
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Abstract. The homotopy category of complexes of projective leftmodules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite derived
Residue complexes over noncommutative rings
 J. ALGEBRA
, 2001
"... Residue complexes were introduced by Grothendieck in algebraic geometry. These are canonical complexes of injective modules that enjoy remarkable functorial properties (traces). In this paper we study residue complexes over noncommutative rings. These objects have a more intricate structure than in ..."
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Cited by 5 (5 self)
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Residue complexes were introduced by Grothendieck in algebraic geometry. These are canonical complexes of injective modules that enjoy remarkable functorial properties (traces). In this paper we study residue complexes over noncommutative rings. These objects have a more intricate structure than in the commutative case, since they are complexes of bimodules. We develop methods to prove uniqueness, existence and functoriality of residue complexes. For a polynomial identity algebra over a field (admitting a noetherian connected filtration) we prove existence of the residue complex and describe its structure in detail.
Rigid Dualizing Complexes over Commutative Rings
 DEPARTMENT OF MATHEMATICS BEN GURION UNIVERSITY, BE’ER
, 2006
"... In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. We obtain many of the important local features of Groth ..."
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Cited by 4 (1 self)
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In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. We obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper [Ye4] these results will be used to construct and study rigid dualizing complexes on schemes.
Finite flat and projective dimension
"... It is surprising but true that if A is a nice enough ring, then any Aleftmodule of finite flat dimension also has finite projective dimension. A result to this effect was proved in [3, prop. 6] under the condition that the number FPD(A) = sup pd M M is an Aleftmodule with pd M < ∞ known as the ..."
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Cited by 3 (1 self)
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It is surprising but true that if A is a nice enough ring, then any Aleftmodule of finite flat dimension also has finite projective dimension. A result to this effect was proved in [3, prop. 6] under the condition that the number FPD(A) = sup pd M M is an Aleftmodule with pd M < ∞ known as the left big finitistic projective dimension of A, is finite. Here pdM denotes the projective dimension of M. Unfortunately, this number is not known to be finite even if A is a finite dimensional algebra over a field, where, indeed, its finiteness is a celebrated conjecture. On the other hand, for such an algebra, finite flat certainly implies finite projective dimension, simply because each flat module is projective. So it seems that there might be results based on other conditions than finiteness of FPD(A). The germ of such a result is in [2, cor. 3.4] which shows that finite flat implies finite projective dimension for a ring which is a homomorphic image of a noetherian commutative Gorenstein ring with finite Krull dimension. Now, for such a ring, FPD(A) is in fact finite, and so formally, [2, cor. 3.4] follows from [3, prop. 6]. However, the method of proof of [2, cor. 3.4] lends itself to generalization, and in this note I will use it to show that finite flat implies finite projective dimension for any rightnoetherian algebra which admits a dualizing complex. This includes finite dimensional algebras, but also more interesting cases such as noetherian complete semilocal PI algebras, and filtered algebras whose associated graded algebras are connected and noetherian, and either PI, graded FBN, or with enough normal elements. Setup. k is a field, and A is a rightnoetherian kalgebra for which there exists a leftnoetherian kalgebra B and a dualizing complex BDA.
Double Extension Regular Algebras of Type (14641
 Department of Mathematics, University of Washington
, 2007
"... Abstract. We construct several families of ArtinSchelter regular algebras of global dimension four using double Ore extension and then prove that all these algebras are strongly noetherian, Auslander regular, Koszul and CohenMacaulay domains. Many regular algebras constructed in the paper are new ..."
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Cited by 3 (1 self)
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Abstract. We construct several families of ArtinSchelter regular algebras of global dimension four using double Ore extension and then prove that all these algebras are strongly noetherian, Auslander regular, Koszul and CohenMacaulay domains. Many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an Ore extension of an ArtinSchelter regular algebra of global dimension three. One of the most important projects in noncommutative algebraic geometry is the classification of noncommutative projective 3spaces, or quantum P 3 s. An algebraic version of this project is the classification of ArtinSchelter regular algebras of global dimension four. There has been extensive research on ArtinSchelter regular